One property of lies, noble or otherwise, is that they are
*falsehoods*; and this presupposes an antecedent notion of truth. One
can question whether there is an antecedent notion of truth in
mathematics, which serves as a common ground upon which to resolve
the debate about whether there are infinite sets. It seems to me to
be very analogous to the debate between constructivists and non-
constructivists, where, also, one has to ask: On what none question-
begging grounds could one possibly resolve the issue. (I am also
reminded of groundless and therefore endless debates over the
centuries about religion---although the debates in math have been
less bloody.)
There may be a sense in which one would sometimes say that the axiom
is antecedently true: Namely, it is true to a certain conception that
we have of an ideal structure that the axioms are intended to make
precise. (This is not really the semantical notion of truth.) But one
can expect agreement about truth in this sense only if there is
agreement about what one wants to axiomatize. When what is being
axiomatized is the theory of transfinite numbers, say, then those who
reject infinite objects from the outset will not be arguing about the
truth of the axioms in this sense. Rather they will be rejecting the
whole enterprise---the very grounds upon which any meaningful
discussion of truth or falsity can take place.
Bill Tait